The Schur Multiplicative and Harmonic Convexities for Three Classes of Symmetric Functions

Abstract

We investigate the Schur harmonic convexity for two classes of symmetric functions and the Schur multiplicative convexity for a class of symmetric functions by using a new method and generalizing previous result. As applications, we establish some inequalities by use of the theory of majorization, in particular, and we give some new geometric inequalities in the -dimensional space.

1. Introduction

Throughout this paper, we denote by the -dimensional Euclidean space, and . For , and , we denote by Furthermore, for , we denote by

Recently, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity for some special functions and their applications have been investigated by many authors; see, for instance, [1–12] and the references therein.

For and , the symmetric function was defined by Guan [13] as where are positive integers. In [13], Guan discussed the Schur convexity and Schur multiplicative convexity for , established some interesting inequalities by use of the theory of majorization, and proposed an open problem as follows.

Open Problem A. If and , then the function is Schur convex, or Schur concave.

Chu et al. [2] solved the Open Problem A and established the following Theorem A.

Theorem A. If and , then the function is Schur concave in and Schur convex in .

Xia and Chu [14] defined the following symmetric function: where , , and are positive integers. In [14], they discussed the Schur convexity and Schur multiplicative convexity for and established some inequalities.

For and , the symmetric function was defined by Xia et al. [15] as where are positive integers. In [15], Xia et al. proved that is Schur concave and Schur harmonic convex in (see [15, Theorems 3.1-3.2]) and Schur multiplicatively concave in , while is Schur multiplicatively convex in , and is Schur multiplicatively concave in (see [15, Theorem 3.3]).

It is natural to ask whether the symmetric functions and are Schur harmonic convex, or Schur harmonic concave in , while the function is Schur multiplicatively convex or Schur multiplicatively concave for and , which have not been studied in existing literatures.

The main purpose of this paper is to discuss the Schur harmonic convexity of and in and solve the problem of the Schur multiplicative convexity or concavity of in by using a new method, the main difficulty in proving Theorem 3 is to apply some techniques of differential calculus and combinatorics, in particular, and construct one complicate equality (29) in proof of Theorem 3. As applications, we establish some inequalities by use of the theory of majorization, in particular, we give some new geometric inequalities in the -dimensional space.

Our main results are the following three theorems.

Theorem 1. For , and .
(1) The function is Schur harmonic convex in ;
(2) If is even integer (or odd integer, respectively), then is Schur harmonic convex (or concave, respectively) in .

Theorem 2. For , and .
(1) The function is Schur harmonic concave in ;
(2) if is even integer (or odd integer, respectively), then is Schur harmonic concave (or convex, respectively) in .

Theorem 3. For , , and , is Schur multiplicatively convex in and Schur multiplicatively concave in .

Remark 4. From the proof of Theorem 3.3 in [15], it is easy to see that Theorem 3 is valid for and , and Theorems 3 is a generalization of the Theorem 3.3 in [15], which is the main result in [15], thus, by using a new method we generalize and solve the problem of the Schur multiplicative convexity or concavity of in from [15].

This paper, except for the introduction, is divided into three sections. In Section 2, we recall some definitions and lemmas. By using the results of Section 2, we give the proof of the main results in Section 3. Finally, some applications are given by use of the theory of majorization.

2. Definitions and Lemmas

For convenience, we recall some definitions.

Definition 5. Let be a set; a real-valued function on is said to be Schur convex if for each pair of -tuples and in , such that , that is and where denotes the th largest component in . is called Schur concave if is Schur convex.

The above notion of Schur convexity was first introduced by Schur in 1923 and has many important applications in analytic inequalities [16–20], isoperimetric problem for polytopes [21], linear regression [22], combinatorial optimization [23], graphs and matrices [24], gamma and digamma functions [25], information-theoretic topics [26], stochastic orderings [27], and other related fields. The Definition 5 can be found in many references such as [20, 28]. Following Schur, the following definition of the Schur multiplicative convex function was introduced in [29, 30].

Definition 6. Let be a set; a real-valued function on is said to be Schur multiplicatively convex if for each pair of -tuples and in , such that . is called Schur multiplicatively concave if is Schur multiplicatively convex.

Recently, Chu and Lv [31] introduced the notion of Schur harmonic convexity.

Definition 7. Let be a set; a real-valued function on is said to be Schur harmonic convex function if for each pair of -tuples and in , such that . is called Schur harmonic concave if inequality (10) is reversed.

In order to establish our main result, we need two lemmas.

Lemma 8 (see [29, 30]). Let be a symmetric multiplicatively convex set with nonempty interior in and be a continuous symmetry function on . If is differentiable in , then is Schur multiplicatively convex on if and only if for all . And is Schur multiplicatively concave on if and only if inequality (11) is reversed. Here is a multiplicatively convex set which means that for .

Lemma 9 (see [31]). Let be a symmetric harmonic convex set with nonempty interior in and be a continuous symmetry function on . If is differentiable on , then is Schur harmonic convex on if and only if for all . And is Schur harmonic concave in if and only if inequality (12) is reversed. Here is a harmonic convex set which means that for .

3. Proofs of the Main Results

Proof of Theorem 1. (1) According to Lemma 9, we only need to prove that for all and . The proof is divided into three cases.
Case 1. If , then from (3) we have By (15), we obtain Case 2. If and , then from (3) we obtain By (18), we obtain Case 3. If and , noting that the r-th order elementary symmetric function (see [32]) is defined as where , , and are positive integers and
Let , and then from (3), we haveFrom (22) and (23), we obtain From Cases 1–3 we obtain that (13) holds for all and ; therefore, is schur harmonic convex in .
(2) Here we give only the proof in the case of being even integer, since the proof in the case of being odd integer is similar. According to Lemma 9, we only need to prove that (13) holds for all . From the proofs of Cases 2-3 in (1), it is easy to see that for ; thus we derive that (13) holds for all in the case of being even integer; therefore, is Schur harmonic convex in .
The proof of Theorem 1 is completed.

Similar to the proof of Theorem 1, we can use Lemma 9 and (20) to prove Theorem 2; therefore we omit the details of the proof. Next we give the proof of Theorem 3.

Proof of Theorem 3. According to Lemma 8, we only need to discuss the nonnegativity and nonpositivity of for all and .
Let ; then from (5) and (20) we have where . From (26) and (27) we obtain where Note that and Let ; then , from (30) we have From (31), we have for ; hence by (33), if , then for ; therefore, is increasing on with ; further from (32) and (34), we obtain for and have for () Hence by (35) and (36), respectively, if , then is decreasing on with , and if , then is increasing on with ; further from (28) we derive that for , and for . Therefore, from Lemma 8, is Schur multiplicatively convex in and Schur multiplicatively concave in .
The proof of Theorem 3 is completed.

Remark 10. It is easy to see that the key point of our proof of Theorem 3 is to construct one monotone function and prove that is decreasing on , and increasing on with , which is different from the proof of Theorem 3.3 in [15].

4. Applications

In this section, we establish some inequalities by use of Theorems 1–3 and the theory of majorization.

Theorem 11. Let , , , and .
(1) Suppose that . If is even integer, then while if is odd integer, then the inequalities (37) and (38) are reversed
(2) If , the inequalities (37) and (38) hold.

Proof. It is easy to see that From Theorems 1 and 2, respectively, and applying (39), we get that Theorem 11 holds.

Theorem 12. Let , , , and .
(1) If , then (2) Suppose that . If is even integer, then the inequalities (40) and (41) hold, while if is odd integer, the inequalities (40) and (41) are reversed.

Proof. It is easy to see that From Theorems 1 and 2, respectively, and applying (42), we obtain that Theorem 12 holds.

Theorem 13. Let , , , and . If , then if , the inequality (43) is reversed.

Proof. It is easy to see that From Theorems 3 and Remark 4, and applying (44), we obtain that Theorem 13 holds.

Remark 14. It is easy to see that Theorem 13 is a generalization of the Theorem 4.5 in [15].

Theorem 15. Let be an -dimensional simples in and be an arbitrary point in the interior of . If is the intersection point of straight and hyperplane . Then for

Proof. It is easy to see that and ; these imply that Therefore, (45)-(48) follow from Theorems 1(2) and 2(2), respectively, together with (49) and (50).

Remark 16. The inequalities (45)-(48) can be found in [33, p.473-480]; here we give a different proof of them.

Theorem 17. Let be an -dimensional simples in and be the inradius of . Let be the altitude of from the vertex , i.e., the distance from to hyperplane , and be the radius of -th escribed hypersphere of . Then for

Proof. Let denotes the volume of , denote the -dimensional volume of the side face of opposite to the vertex (i=1,2,…,n+1). Let ; noting the well-known fact (see [33, p.463]) we obtain Thus we have and further we obtain Therefore, (51)-(54) follow from Theorems 1(2) and 2(2), respectively, together with (58) and (59).

Remark 18. The inequality (51) can be found in the Proposition 4 from [19]; here we give a different proof of it. Mitrinović, Pečarić, and Volenec (see [33, p.463-473]) established a series of inequalities for , , and ; obviously, our inequalities (52)-(54) in Theorem 17 are different from theirs.

5. Conclusions

In this paper, we investigate the Schur harmonic convexity for two classes of symmetric functions defined by Guan (2007), Xia and Chu (2009), and the Schur multiplicative convexity for a class of symmetric functions defined by Xia et al. (2010) by using a new method and generalize the main results of Xia et al. (2010). As applications, we establish some inequalities by use of the theory of majorization; in particular, we give some new geometric inequalities in the -dimensional space. It is easy to see that

(1) Our Theorem 13 is a generalization of the Theorem 4.5 in Xia et al. (2010).

(2) The inequalities (45)-(48) can be found in Mitrinović (1989); here we give a different proof of them.

(3) The inequality (51) can be found in the Proposition 4 from Wu (2005); here we give a different proof of it.

Data Availability

No data of any type is used in this article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported in part by Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, Nature Science Foundation of Hunan Province (Grant no. 12JJ3002), and National Nature Science Foundation of China (Grant no. 11271118).